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SCIENCE 



[N. S. Vol. XXVIII. No. 713 



vanced. work in mathematics. It may not 

 be out of place, therefore, for me to call 

 attention to the fact that in the Massa- 

 chusetts Institute of Technology we have 

 offered and given, among others, the fol- 

 lowing courses: advanced calculus, vector 

 analysis, fourier series, least squares, the- 

 ory of surfaces, theory of functions, el- 

 liptic functions, hydrodynamics and dif- 

 ferential equations of mechanics and phys- 

 ics. Some of these subjects are required 

 in one or more of our courses, but not in 

 any one of the larger engineering courses, 

 which are taken as the basis of Professor 

 Townsend's tables. This elective work, 

 therefore, while valuable in many respects, 

 is not the main work of the mathematical 

 department. 



The mathematical teacher is in the engi- 

 neering school primarily to teach to stu- 

 dents of engineering the amount of mathe- 

 matics which is necessary to them for the 

 proper understanding and practise of their 

 profession. The object is to give the stu- 

 dent a grasp of mathematical concepts and 

 processes through their use, as one learns 

 grammar by speaking a language. Hence 

 there is no place in the required mathe- 

 matics of a technical school, nor indeed in 

 the first courses in a college of liberal arts, 

 for the refinements of modern "rigor." 

 At the same time there should be no pa- 

 tience with a loose or unscientific presen- 

 tation of first principles. The teacher him- 

 self must be thoroughly conversant with 

 modern thought, else he will teach false- 

 hood for truth, and must be enthusiastic in 

 his interest in his subject, else he will fail 

 to inspire his pupils. Hence the teacher 

 of mathematics should be primarily a 

 mathematician and not an engineer. It is 

 hard to find an engineer who has any 

 knowledge of mathematics other than a 

 small fragment which he habitually uses, 

 and any elementary teacher whose instruc- 

 tion goes to the very limits of his knowledge 



is sure of failure. It may, of course, be 

 possible to superimpose a mathematical 

 training upon an engineering one, but iu 

 that case the engineer becomes a mathe- 

 matician and my contention that mathe- 

 matics should be taught by a mathema- 

 tician is not invalidated. 



On the other hand, the mathematician 

 should know something of the uses to which 

 an engineer wishes to put mathematics. 

 For that reason such meetings as this are 

 helpful, but I must confess to feeling a 

 little disappointment in not obtaining from 

 the engineers any new light on the concrete 

 problem which confronts the teacher of 

 mathematics in an engineering school. I 

 have met the same disappointment else- 

 where in similar meetings. It has hap- 

 pened, elsewhere if not here, that engineers 

 will tell the mathematicians what and how 

 they should teach, in apparently total ig- 

 norance of the fact that what the engineer 

 promulgates as a new gospel has been the 

 commonplace thought of the mathematician 

 for years. This ignorance may be due to 

 the fact that the engineer remembers his 

 own training of twenty or thirty years ago 

 and does not know that improvements 

 have taken place. That such is the 

 case may be seen by a comparison 

 of modern with older text-books. Such 

 criticism from the engineers is amusing, 

 but another kind of criticism is not. I 

 refer to the kind which seizes upon the 

 failure of a student to have learned mathe- 

 matics thoroughly as evidence of poor aims 

 and inefficient teaching of the mathemat- 

 ical instructor. "We all know that students 

 pass through our classes and graduate from 

 our schools whose attainments are not what 

 we wish, but while the mathematical 

 teacher delivers his product to the engi- 

 neering departments and hears of his com- 

 parative failures, the engineering professor 

 delivers his product to the world and rarely 

 hears of the specific blunders of his stu- 



